Information processing apparatus and method

ABSTRACT

An information processing method for system identification includes: generating a fitting curve represented by a sum of exponential functions for each of a set of digital inputs and a set of digital outputs for a physical system that is represented by one or plural equations including m-order differential operators (m is an integer equal to or greater than 1); and calculating coefficients of the differential operators, which are included in first coefficients, so that a first coefficient of each exponential function included in an expression obtained by a product of the differential operators and the fitting curve for the set of the digital inputs is equal to a second coefficient of the same exponential function, which is included in the fitting curve for the set of the digital outputs.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is based upon and claims the benefit of priority of theprior Japanese Patent Application No. 2013-264613, filed on Dec. 20,2013, and the Japanese Patent Application No. 2014-189103, filed on Sep.17, 2014, the entire contents of which are incorporated herein byreference.

FIELD

This invention relates to a system identification technique.

BACKGROUND

In a field such as an automatic control, there is an issue ofidentification of a linear system. For example, when an analogue inputu(t) and an analogue output y(t) are given with respect to a physicalsystem, which is represented by a linear ordinary differential equationas illustrated in FIG. 1, a problem is to determine differentialcoefficients P₁ to P_(m) of differential operators P(d/dt) so that thefollowing relations hold.

${{P\left( \frac{}{t} \right)}{y(t)}} = {u(t)}$${P\left( \frac{}{t} \right)} = {\left( \frac{}{t} \right)^{m} + {P_{1}\left( \frac{}{t} \right)}^{m - 1} + \ldots + {P_{m - 1}\left( \frac{}{t} \right)} + P_{m}}$

In this patent application, a case where P(d/dt) y(t) is almost equal tou(t) is represented as P(d/dt)y(t) being equal to u(t).

However, sensor values such as output values of an accelerometer, whichis recently used, are digital values (i.e. discrete values) instead ofanalogue values. Therefore, as illustrated in FIG. 2, when digitalinputs u₀, u₁, u_(2n) and digital outputs y₀, y₁, . . . , y_(2n) aregiven, a method for determining differential coefficients P₁ to P_(m) ofthe differential operators P(d/dt) is considered so that thedifferential equation P(d/dt)y(t)=u(t) of the physical system, which isrepresented by the ordinary differential equation, holds.

Therefore, in order to realize this, the digital data is converted tothe analogue data. As for that method, there is a fitting method. Thespline method is a conventional famous fitting method, however, it has aproblem that it is possible to differentiate the spline curve a fewtimes, for example. Therefore, when order m of the differential operatoris high, the spline method cannot be applied.

Moreover, even when another fitting method is adopted, it is notrealistic in view of the calculation amount that the differentialoperations are calculated for all times, and it is not possible toidentify the system that is represented by the differential equationincluding higher-order differential operators.

-   Patent Document 1: Japanese Laid-open Patent Publication No.    2013-175143-   Patent Document 2: Japanese Laid-open Patent Publication No.    2010-264499-   Patent Document 3: Japanese Laid-open Patent Publication No.    2000-182552-   Patent Document 4: Japanese Laid-open Patent Publication No.    2003-108542-   Patent Document 5: Japanese Laid-open Patent Publication No.    2007-286801-   Patent Document 6: Japanese Laid-open Patent Publication No.    2001-208842-   Non-Patent Document 1: T. Ito, Y. Senta and F. Nagashima, “Analyzing    Bilinear Neural Networks with New Curve Fitting for Application to    Human Motion Analysis”, in Proc. IEEE Int. Conf. Systems, Man and    Cybernetics, 2012, pp. 345-352

SUMMARY

An information processing method relating to this invention includes (A)generating a fitting curve represented by a sum of exponential functionsfor each of a set of digital inputs and a set of digital outputs for aphysical system that is represented by one or plural equations includingm-order differential operators (m is an integer equal to or greater than1.) The set of the digital inputs and the set of the digital outputs arestored in a data storage unit; and (B) calculating coefficients of thedifferential operators, which are included in first coefficients, sothat a first coefficient of each exponential function included in anexpression obtained by a product of the differential operators and thefitting curve for the set of the digital inputs is equal to a secondcoefficient of the same exponential function, which is included in thefitting curve for the set of the digital outputs.

The object and advantages of the embodiment will be realized andattained by means of the elements and combinations particularly pointedout in the claims.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory and arenot restrictive of the embodiment, as claimed.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a diagram schematically depicting a system of analog inputsand outputs;

FIG. 2 is a diagram schematically depicting a system of digital inputsand outputs;

FIG. 3 is a diagram depicting an outline of a system relating toembodiments;

FIG. 4 is a functional block diagram of an information processingapparatus relating to the embodiments;

FIG. 5 is a diagram depicting a processing flow of a main processingrelating to a first embodiment;

FIG. 6 is a diagram schematically depicting a system of the digitalinputs and outputs;

FIG. 7 is a diagram depicting a processing flow of a learningprocessing;

FIG. 8 is a diagram to explain a linear regression;

FIG. 9 is a diagram depicting a processing flow of a predictionprocessing;

FIG. 10 is a diagram schematically depicting a system of digital inputsand outputs;

FIG. 11 is a diagram schematically depicting a system of digital inputsand outputs in a second embodiment;

FIG. 12 is a diagram depicting a processing flow of a main processingrelating to the second embodiment; and

FIG. 13 is a functional block diagram of a computer.

DESCRIPTION OF EMBODIMENTS Embodiment 1

FIG. 3 illustrates an example of a system relating to this embodiment.For example, a detection apparatus 100 is provided in a vehicle 1000,and the detection apparatus 100 has an accelerometer 101 that measuresacceleration of the vehicle 1000, a distance sensor 102 that measures arunning distance of the vehicle 1000 and a communication unit 103.

The communication unit 103 of the detection apparatus 100 in the vehicle1000 transmits measurement data to an information processing apparatus300, for example, within a cloud, data center or the like through anetwork 200 such as the Internet. The vehicle 1000 is a physical systemthat is represented by differential equations including differentialoperators of m-order (m is an integer that is equal to or greaterthan 1) as illustrated in FIG. 2. Moreover, the vehicles 1000 may beplural instead of single.

FIG. 4 illustrates a configuration example of the information processingapparatus 300.

The information processing apparatus 300 has a communication unit 310,an input/output data storage unit 320, a learning processing unit 330and a prediction processing unit 340.

The communication unit 310 receives measurement data from the detectionapparatus 100 in the vehicle 1000, and stores the received measurementdata in the input/output data storage unit 320. The learning processingunit 330 performs a processing to identify the differential equationsrepresenting the vehicle 1000 from the digital inputs and digitaloutputs, and has a first fitting curve generator 331, a first datastorage unit 332, a first operation unit 333 and a second data storageunit 334.

The first fitting curve generator 331 performs a Discrete FourierTransformation (DFT) to generate a fitting curve for the digital inputsthat are stored in the input/output data storage unit 320 and a fittingcurve for the digital outputs that are stored in the input/output datastorage unit 320, and stores data of those curves in the first datastorage unit 332.

The first operation unit 333 has a first coefficient calculation unit3331 and a linear regression processing unit 3332. The first coefficientcalculation unit 3331 generates data used in the linear regressionprocessing unit 3332 from data of the fitting curves, which is stored inthe first data storage unit 332, and stores the generated data in thefirst data storage unit 332. The linear regression processing unit 3332performs a linear regression processing by using data stored in thefirst data storage unit 332, and stores processing results in the seconddata storage unit 334. The processing results include the differentialcoefficients of the m-order differential operators. When thedifferential coefficients of the m-order differential operators areobtained, the differential equations that represent the vehicle 1000will be obtained.

Moreover, the prediction processing unit 340 performs a processing topredict the digital outputs from new digital inputs and the identifieddifferential equations, and has a second fitting curve generator 341, athird data storage unit 342, a second operation unit 343 and a fourthdata storage unit 344.

The second fitting curve generator 341 performs a similar processing tothe first fitting curve generator 331 to generate a fitting curve forthe digital inputs stored in the input/output data storage unit 320, andstores data of that curve in the third data storage unit 342.

The second operation unit 343 has a second coefficient calculation unit3431 and an output data calculation unit 3432. The second coefficientcalculation unit 3431 calculates coefficients of an output function fromthe coefficients of the differential operators, which are stored in thesecond data storage unit 334, and data of the fitting curves, which isstored in the third data storage unit 342. The output data calculationunit 3432 calculates output data by the output function identified withthe coefficients calculated by the second coefficient calculation unit3431, and stores the output data in the fourth data storage unit 344.

Next, processing contents of the information processing apparatus 300will be explained. The learning processing unit 330 of the informationprocessing apparatus 300 performs a learning processing that will beexplained later (FIG. 5: step S1). The learning processing will beexplained by using FIGS. 6 to 8 in detail. After that, the predictionprocessing unit 340 performs a prediction processing based on thelearning result of the learning processing unit 330 (step S3). Theprediction processing will be explained by using FIG. 9 in detail.

Next, the learning processing will be explained by using FIGS. 6 to 8.It is assumed that the communication unit 103 of the detection apparatus100 in the vehicle 1000 transmits data of the digital inputs (e.g.measurement data of the accelerometer 101) and the digital outputs (e.g.measurement data of the distance sensor 102) at arbitrary timings, andthe communication unit 310 of the information processing apparatus 300stores received data from the vehicle 1000 in the input/output datastorage unit 320.

More specifically, as illustrated in FIG. 6, it is assumed that outputsY_(j) (j=1 to l) of the system are obtained for the digital inputsU_(j). U_(j) and Y_(j) include 2n+1 discrete points as described later.

U _(j)=(u _(j,0) ,u _(j,1) , . . . ,u _(j,2n−1) ,u _(j,2n))j=1,2 . . . l

Y _(j)=(y _(j,0) ,y _(j,1) , . . . ,y _(j,2n−1) ,y _(j,2n))j=1,2 . . . l

As described above, this system is represented by the differentialequation P(d/dt)y_(j) (t)=u_(j)(t) including the m-order differentialoperators P(d/dt).

Then, the first fitting curve generator 331 of the learning processingunit 330 reads out U_(j) and Y_(j) from the input/output data storageunit 320 (FIG. 7: step S11). Then, the first fitting curve generator 331generates a fitting curve u_(j)(t) for the digital inputs U₃ and afitting curve y_(j)(t) for the digital outputs Y_(j) by performing thefitting processing by the DFT for each j, and stores data of the curvesin the first data storage unit 332 (step S13).

This fitting processing will be explained by using the digital outputsY_(j). However, the similar processing will be performed for the digitalinputs U_(j). Moreover, in order to make it easy to understand theexplanation, “j” is omitted.

More specifically, the DFT is performed for real values y₀, y₁, . . . ,y_(2n) of the discrete points to calculate n+1 frequencies. In otherwords, q=0, 1, . . . , n.

$Y_{q} = {{\sum\limits_{i = 0}^{2n}{y_{i}^{\frac{{- 2}\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}}} = {\sum\limits_{i = 0}^{2n}{y_{i}\left( {{\cos \frac{2{qi}\; \pi}{{2n} + 1}} - {\sqrt{- 1}\sin \frac{2{qi}\; \pi}{{2n} + 1}}} \right)}}}$

Each frequency Y₀, Y₁, . . . , Y_(n) is complex number, and a real parta_(q) and an imaginary part b_(q) are represented as follows:

Y ₀ =a ₀(b ₀=0)

Y _(q) =a _(q)−√{square root over (−1)}b _(q) q=0,1, . . . ,n

Because the discrete points are real values, please note b₀=0 from thecharacteristic of the DFT.

By using a_(q) and b_(q) obtained as described above, finite (i.e.n-order) Fourier series are generated as described below.

$\begin{matrix}{{y(t)} = {{\frac{1}{{2n} + 1}a_{0}} + {\frac{2}{{2n} + 1}{\sum\limits_{q = 1}^{n}\left( {{a_{q}\cos \frac{2q\; \pi}{{2n} + 1}t} + {b_{q}\sin \frac{2q\; \pi}{{2n} + 1}t}} \right)}}}} & (1)\end{matrix}$

The Fourier series obtained here are the fitting curve of the discretepoints y₀, y₁, . . . , y_(n). This fitting curve passes through thesediscrete points as will be explained later.

When 2n+1 discrete points y₀, y₁, . . . , y_(2n) are obtained, it ispossible to obtain 2n+1 frequencies by the DFT.

$Y_{q} = {{\sum\limits_{i = 0}^{2n}{y_{i}^{\frac{{- 2}\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}}} = {\sum\limits_{i = 0}^{2n}{y_{i}\left( {{\cos \frac{2{qi}\; \pi}{{2n} + 1}} - {\sqrt{- 1}\sin \frac{2{qi}\; \pi}{{2n} + 1}}} \right)}}}$q = 0, 1, …  , 2n

When the discrete points are real values, following relations areobtained.

Y _(q) = Y _(2n+l−q)

Y with a bar represents the complex conjugation of Y. According to this,following relations are also obtained.

a _(q) =a _(2n+1−q) b _(q) =−b _(2n+1−q) q=0,1, . . . ,2n

Here, when an Inverse Discrete Fourier Transformation (IDFT) isperformed in order to recover data of original discrete points from thefrequencies Y₀, Y₁, . . . , Y_(2n), the IDFT is represented like a firstline in the following expression.

$\begin{matrix}{{\left( {{2n} + 1} \right)y_{i}} = {\sum\limits_{q = 0}^{2n}{Y_{q}^{\frac{2\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}}}} \\{= {Y_{0} + {\sum\limits_{q = 1}^{n}{Y_{q}^{\frac{2\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}}} + {\sum\limits_{q = {n + 1}}^{2n}{{\overset{\_}{Y}}_{{2n} + 1 - q}^{\frac{2\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}}}}} \\{= {Y_{0} + {\sum\limits_{q = 1}^{n}{Y_{q}^{\frac{2\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}}} + {\sum\limits_{k = 1}^{n}{{\overset{\_}{Y}}_{k}^{\frac{2\sqrt{- 1}{({{2n} + 1 - k})}i\; \pi}{{2n} + 1}}}}}} \\{= {Y_{0} + {\sum\limits_{q = 1}^{n}\left( {{Y_{q}^{\frac{2\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}} + {{\overset{\_}{Y}}_{q}^{\frac{{- 2}\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}}} \right)}}} \\{= {a_{0} + {\sum\limits_{q = 1}^{n}\begin{pmatrix}{{\left( {a_{q} - {\sqrt{- 1}b_{q}}} \right)^{\frac{2\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}} +} \\{\left( {a_{q} - {\sqrt{- 1}b_{q}}} \right)^{\frac{{- 2}\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}}\end{pmatrix}}}} \\{= {a_{0} + {\sum\limits_{q = 1}^{n}\begin{pmatrix}{{a_{q}\left( {^{\frac{2\sqrt{- 1}{qi}\; \pi}{{2n} + 1}} + ^{\frac{{- 2}\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}} \right)} -} \\{\sqrt{- 1}{b_{q}\left( {^{\frac{2\sqrt{- 1}{qi}\; \pi}{{2n} + 1}} - ^{\frac{{- 2}\sqrt{- 1}{qi}\; \pi}{{2n} + 1}}} \right)}}\end{pmatrix}}}} \\{= {a_{0} + {2{\sum\limits_{q = 1}^{n}\left( {{a_{q}{\cos \left( \frac{2{qi}\; \pi}{{2n} + 1} \right)}} + {b_{q}{\sin \left( \frac{2{qi}\; \pi}{{2n} + 1} \right)}}} \right)}}}}\end{matrix}$ i = 0, 1, …  , 2n

This second line represents that the right side of the first line isdivided according to the aforementioned relationship between Y and itscomplex conjugation. In third and subsequent lines, a parameter q isadjusted and arrangement is advanced by representing the expression withthe real part a_(q) and imaginary part b_(q) of the frequencies, andthen, the last line is obtained.

On the other hand, when n-order Fourier series y(t), which arerepresented by the expression (1), are sampled at t=0, 1, . . . , 2n, afollowing expression is obtained.

$\begin{matrix}{{{\left( {{2n} + 1} \right){y_{i}(i)}} = {a_{0} + {2{\sum\limits_{q = 1}^{n}\left( {{a_{q}{\cos \left( \frac{2{qi}\; \pi}{{2n} + 1} \right)}} + {b_{q}{\sin \left( \frac{2{qi}\; \pi}{{2n} + 1} \right)}}} \right)}}}}{{i = 0},1,\ldots \mspace{14mu},{2n}}} & (2)\end{matrix}$

Because the right side of this expression (2) is equal to the last lineof the expression of the Inverse Discrete Fourier Transformation, it isunderstood that y_(i)=y(i) (i=0, 1, . . . , 2n) holds. In other words,it is understood that the n-order Fourier series y(t) in the expression(1) pass through each of the original discrete points.

Moreover, when arbitrary n-order Fourier series g(t) are given andsampled at t=0, 1, . . . , 2n to obtain discrete points g_(i)=g(i) (i=0,1, . . . , 2n), the fitting curve of the discrete points g_(i) (i=0, 1,. . . , 2n) is calculated, and the original g(t) is obtained again. Thiscan be proved when a similar method to the aforementioned method isused, however, the detailed explanation is omitted here. As describedabove, this fitting method realizes one-to-one correspondence betweenthe discrete points and finite Fourier series.

Therefore, the fitting curve y_(j)(t) for the digital outputs and thefitting curve u_(j)(t) for the digital inputs are obtained as describedbelow.

${y_{j}(t)} = {{\frac{1}{{2n} + 1}a_{j,0}} + {\frac{2}{{2n} + 1}{\sum\limits_{q = 1}^{n}\left( {{a_{j,q}\cos \frac{2q\; \pi}{{2n} + 1}t} + {b_{j,q}\sin \frac{2q\; \pi}{{2n} + 1}t}} \right)}}}$j = 1, 2  …  l${u_{j}(t)} = {{\frac{1}{{2n} + 1}{\overset{\sim}{a}}_{j,0}} + {\frac{2}{{2n} + 1}{\sum\limits_{q = 1}^{n}\left( {{{\overset{\sim}{a}}_{j,q}\cos \frac{2q\; \pi}{{2n} + 1}t} + {{\overset{\sim}{b}}_{j,q}\sin \frac{2q\; \pi}{{2n} + 1}t}} \right)}}}$j = 1, 2  …  l

Moreover, this fitting curve y_(j)(t) and u_(j)(t) can be represented bya sum of exponential functions whose base is the Napier's constant e.When the fitting curves can be represented by the sum of the exponentialfunctions, it is possible to easily obtain the m-order differential.

$\begin{matrix}\begin{matrix}{{y_{j}(t)} = {{\frac{1}{{2n} + 1}a_{j,0}} + {\frac{2}{{2n} + 1}{\sum\limits_{q = 1}^{n}\left( {{a_{j,q}\cos \frac{2q\; \pi}{{2n} + 1}t} + {b_{j,q}\sin \frac{2q\; \pi}{{2n} + 1}t}} \right)}}}} \\{= {\sum\limits_{q = {- n}}^{n}{c_{j,q}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}}}\end{matrix} & (3) \\\begin{matrix}{{u_{j}(t)} = {{\frac{1}{{2n} + 1}{\overset{\sim}{a}}_{j,0}} + {\frac{2}{{2n} + 1}{\sum\limits_{q = 1}^{n}\left( {{{\overset{\sim}{a}}_{j,q}\cos \frac{2q\; \pi}{{2n} + 1}t} + {{\overset{\sim}{b}}_{j,q}\sin \frac{2q\; \pi}{{2n} + 1}t}} \right)}}}} \\{= {\sum\limits_{q = {- n}}^{n}{d_{j,q}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}}}\end{matrix} & \;\end{matrix}$

c_(j,q) and d_(j,q) are represented as follows:

$\begin{matrix}{c_{j,q} = {{\frac{a_{j,q} - {\sqrt{- 1}b_{j,q}}}{{2n} + 1}\left( {q > 0} \right)c_{j,q}} = {{\frac{a_{j,q} - {\sqrt{- 1}b_{j,q}}}{{2n} + 1}\left( {q < 0} \right)c_{j,0}} = \frac{a_{j,0}}{{2n} + 1}}}} & (4) \\{d_{j,q} = {{\frac{{\overset{\sim}{a}}_{j,q} - {\sqrt{- 1}{\overset{\sim}{b}}_{j,q}}}{{2n} + 1}\left( {q > 0} \right)d_{j,q}} = {{\frac{{\overset{\sim}{a}}_{j,q} - {\sqrt{- 1}{\overset{\sim}{b}}_{j,q}}}{{2n} + 1}\left( {q < 0} \right)d_{j,0}} = \frac{{\overset{\sim}{a}}_{j,0}}{{2n} + 1}}}} & \;\end{matrix}$

In this embodiment, the first fitting curve generator 331 generates afitting curve by calculating coefficients c_(j,q) and d_(j,q) for each j(j=1 to 1) and each q (q=−n to +n) according to the expressions (3) and(4). Those c_(j,q) and d_(j,q) are stored in the first data storage unit332, for example.

Then, the first coefficient calculation unit 3331 of the first operationunit 333 generates plural points (C, D) on an αβ plane for calculatingeach differential coefficient of the m-th differential operator from thecoefficients c_(j,q) and d_(j,q) (step S15).

As described above, because of P(d/dt)y_(j)(t)=u_(j) (t), P(d/dt)y_(j)(t) on the left side is represented as follows:

$\begin{matrix}{{{P\left( \frac{}{t} \right)}{y_{j}(t)}} = {\sum\limits_{q = {- n}}^{n}{c_{j,q}{P\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}}} & (5) \\{{P(x)} = {x^{m} + {P_{1}x^{m - 1}} + \ldots + {P_{m - 1}x} + P_{m}}} & \; \\{{j = 1},{2\mspace{14mu} \ldots \mspace{14mu} l}} & \;\end{matrix}$

As described above, y_(j) is a sum of the exponential functions.Therefore, in case of the first-order differentiation, its coefficientbecomes a coefficient of the exponent of the exponential function, incase of the second-order differential, its coefficient becomes (thecoefficient of the exponent of the exponential function)², in case ofthe third-order differential, its coefficient becomes (the coefficientof the exponent of the exponential function)³, and in case of them-order differential, its coefficient becomes (the coefficient of theexponent of the exponential function)^(m). Thus, the differentialoperation becomes very easy.

In other words, the expression is represented as follows:

$\begin{matrix}{{{P\left( \frac{}{t} \right)}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}} = {{\left( \frac{}{t} \right)^{m}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}} + {{P_{1}\left( \frac{}{t} \right)}^{m - 1}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}} + \ldots +}} \\{{{P_{m - 1}\frac{}{t}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}} + {P_{m}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}}} \\{= {\begin{Bmatrix}{\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)^{m} + {P_{1}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{m - 1} + \ldots +} \\{{P_{m - 1}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)} + P_{m}}\end{Bmatrix}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}} \\{= {{P\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}}\end{matrix}$

Therefore, the aforementioned expression (5) is obtained.

Then, P(d/dt)y_(j)(t)=u_(j)(t) is represented as follows:

$\begin{matrix}{{{\sum\limits_{q = {- n}}^{n}{c_{j,q}{P\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}} = {\sum\limits_{q = {- n}}^{n}{d_{j,q}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}}}{{j = 1},{2\mspace{14mu} \ldots \mspace{14mu} l}}} & (6)\end{matrix}$

Because each exponential function in the expression (6) is orthogonal,when the coefficient of each exponential function on the left side isidentical to the coefficient of the same exponential function on theright side, the entire of the left side is identical to the entire ofthe right side. In other words, following relations are obtained.

$\begin{matrix}{{{{c_{j,q}{P\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}} = {{{c_{j,q}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{m} + {{c_{j,q}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{m - 1}P_{1}} + \ldots + {c_{j,q}P_{m}}} = d_{j,q}}}\mspace{20mu} {{j = 1},{2\mspace{14mu} \ldots \mspace{14mu} l}}}\mspace{20mu} {{q = {- n}},\ldots \mspace{14mu},n}} & (7)\end{matrix}$

Here, because d_(j,q) and c_(j,q)(2√(−1)qπ/(2n+1))^(m) are constant,they are moved to the right side, and terms of unknown differentialcoefficients P₁ to P_(m) are moved to the left side. Therefore, afollowing expression is obtained.

$\begin{matrix}{{{{{{c_{j,q}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{m - 1}P_{1}} + \ldots + {{c_{j,q}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}P_{m - 1}} + {c_{j,q}P_{m}}} = {d_{j,q} - {c_{j,q}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{m}}}\mspace{20mu} {{j = 1},{2\mspace{14mu} \ldots \mspace{14mu} l}}}\mspace{20mu} {{q = {- n}},\ldots \mspace{14mu},n}} & (8)\end{matrix}$

At the step S15, coefficients that are multiplied to unknowndifferential coefficients in this expression (8) are represented as am-dimensional vector C_(j,q), and a value of the constant term in theexpression (8) is represented as D_(j,q) as follows:

$C_{j,q} - \left( {{c_{j,q}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{m - 1},{c_{j,q}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{m - 2},\ldots \mspace{14mu},{c_{j,q}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)},c_{j,q}} \right)$$\mspace{20mu} {D_{j,q} = {d_{j,q} - {c_{j,q}\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}^{m}}}$  j = 1, 2  …  l   q = −n, …  , n

Then, the linear regression processing unit 3332 calculates thedifferential coefficients P₁ to P_(m) by performing the linearregression by the method of least squares for such points (C_(j,q),D_(j,q)), and stores the differential coefficients in the second datastorage unit 334 (step S17).

At this step, the unknown coefficients P₁ to P_(m) of the regressionline β=P(α)=α₁P₁+α₂P₂+ . . . +α_(m-1)P_(m-1)+α_(m)P_(m) are calculated.In other words, as schematically in FIG. 8, when the points explained atthe step S17 are arranged on the αβ plane, the coefficients P₁ to P_(m)that are included in β=P(α) for which the error is minimized arecalculated. The method of least squares is well-known, therefore furtherexplanation is omitted.

By performing the aforementioned processing, the differentialcoefficients of the differential operators are calculated so that theerror is minimized and P(d/dt)y_(j)(t)≈u_(j)(t) is obtained.

As described above, the fitting curve is represented by the sum of theexponential functions. Therefore, it is possible to easily perform thehigher-order differential, and it is also possible to perform theaforementioned processing at high speed.

Next, by using FIG. 9, the processing contents of the predictionprocessing will be explained.

After the learning processing is performed, the detection apparatus 100of the vehicle 1000 transmits only measurement data of the accelerometer101, in other words, data of the digital inputs from the communicationunit 103 to the information processing apparatus 300. Accordingly, atransmission data amount to the information processing apparatus 300 isreduced. Here, instead of the digital inputs used in the learningprocessing, the measurement data of the accelerometer 101 as otherdigital inputs is newly obtained.

Therefore, when the communication unit 310 of the information processingapparatus 300 receives data of the digital inputs from the vehicle 1000,the communication unit 310 stores the received data in the input/outputdata storage unit 320.

Then, the second fitting curve generator 341 of the predictionprocessing unit 340 reads out the data of the digital inputs U (=(u₀,u₁, . . . , u_(2n))) from the input/output data storage unit 320 (FIG.9: step S21). Then, the second fitting curve generator 341 generates afitting curve u(t) for the digital inputs U by the DFT, and stores dataof the curve in the third data storage unit 342 (step S23). d_(q) (q is−n to +n) in the expression (3) is calculated. Because there is onlyj=1, the suffix j is omitted.

Then, the second coefficient calculation unit 3431 of the secondoperation unit 343 uses the results d_(q) of the fitting processing andthe differential coefficients P₁ to P_(m) of the differential operators,which are stored in the second data storage unit 334, to calculatecoefficients c_(q) (q is −n to +n) of the output function y(t) (stepS25).

Specifically, c_(q) is calculated by a following expression.

$c_{q} = \frac{d_{q}}{P\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)}$

Then, the output function y(t) is represented from the coefficientsc_(q) as follows:

$\begin{matrix}{{y(t)} = {\sum\limits_{q = {- n}}^{n}{c_{q}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}}} & (9)\end{matrix}$

Therefore, the output data calculation unit 3432 calculates output dataY=(y₀, y₁, . . . , y_(2n)) for predetermined t (=0, 1, . . . , 2n)according to the expression (9), and stores the output data in thefourth data storage unit 344 (step S27).

Thus, it is possible to estimate a running distance by using thecoefficients of the differential operators, which are obtained by thelearning processing, and to output the estimated running distance.

Embodiment 2

For example, as illustrated in FIG. 10, a case is considered that asystem identification is performed when digital inputs u₀, u₁, . . . ,u_(2m) and digital outputs v₀, v₁, . . . , v_(2m) are obtained.According to the first embodiment, a system 1001 in FIG. 10 isrepresented by using l-order differential operators P(d/dt) as follows:

${{P\left( \frac{}{t} \right)}{u(t)}} = {v(t)}$${P\left( \frac{}{t} \right)} = {\left( \frac{}{t} \right)^{l} + {P_{1}\left( \frac{}{t} \right)}^{l - 1} + \ldots + {P_{l - 1}\left( \frac{}{t} \right)} + P_{l}}$

Here, u(t) is a fitting curve for the digital inputs u₀, u₁, . . . ,u_(2m). v(t) is a fitting curve for the digital outputs v₀, v₁, . . . ,v_(2m). In the second embodiment, in order to make it easy to understandthe explanation, the suffix j of u_(j)(t), v_(j)(t) and y_(j)(t) isomitted.

However, in the first embodiment, it is presumed that the number ofdiscrete points included in the digital inputs and the number ofdiscrete points included in the digital outputs are identical. In otherwords, it is presumed that the sampling intervals of the digital inputsand the sampling intervals of the digital outputs are identical in asection from t=0 to t=2m.

On the other hand, in this embodiment, a conversion system 1101 asillustrated in FIG. 11 is introduced. The conversion system 1101illustrated in FIG. 11 is a system to generate data of 2m+1 discretepoints u₀, u₁, . . . , u_(2m) from the digital inputs y₀, y₁, . . . ,y_(2n). When the conversion system 1101 is introduced, it is possible toperform the system identification from the digital inputs y₀, y₁, . . ., y_(2n) and the digital outputs v₀, v₁, . . . , v_(2m), even if n≠m.

Moreover, the conversion system 1101 is a system that not only canchange the sampling intervals (i.e. change the number of discretepoints) but also can change a type of data by the differential andintegral (e.g. data of the speed is changed to data of the accelerationor data of the distance.). Then, in the second embodiment, theconversion system 1101 is defined by a following expression.

$\begin{matrix}{{{u(t)} = {\frac{^{p}}{t^{p}}{y(t)}\left( {{p = 0},1,\ldots} \right)}}{{u(t)} = {\int{{y(t)}{t}}}}} & (10)\end{matrix}$

y(t) is a fitting curve for the digital inputs y₀, y₁, . . . , y_(2n).In this matter, the conversion system 1101 is a system to generate acurve u(t) by the differential and integral of the fitting curve y(t).When the fitting curve y(t) is generated by the method explained in thefirst embodiment, the differential and integral can be performed,easily. Specifically, the higher-order differential is possible, on theother hand, it is impossible when using the spline method. In case ofp=0, u(t)=y(t) holds.

A system configuration to realize the conversion system 1101 is the sameas that of the first embodiment. In the following, processing contentsof the information processing apparatus 300 in the second embodimentwill be explained by using FIG. 12. This processing is performed beforethe processing in FIG. 5, for example.

The communication unit 103 of the detection apparatus 100 in the vehicle1000 transmits data of digital inputs (e.g. measurement data of theaccelerometer 101) and digital outputs (e.g. measurement data of thedistance sensor 102) to the information processing apparatus 300 atarbitrary timings. The communication unit 310 of the informationprocessing apparatus 300 stores received data from the vehicle 1000 inthe input/output data storage unit 320.

In the second embodiment, it is assumed that y₀, y₁, . . . , y_(2n) arereceived as the digital inputs, and v₀, v₁, . . . , v_(2m) are receivedas the digital outputs. Here, a case of n≠m may occur instead of n=m.

Then, the first fitting curve generator 331 in the learning processingunit 330 reads out the digital inputs y₀, y₁, . . . , y_(2n) from theinput/output data storage unit 320 (FIG. 12: step S31).

The first fitting curve generator 331 generates a fitting curve y(t) forthe digital inputs y₀ y₁, . . . , y_(2n) by performing the fittingprocessing by the DFT (step S33), and stores data of the fitting curvey(t) in the first data storage unit 332. At the step S33, the firstfitting curve generator 331 performs the fitting processing by the DFTfor each j, similarly to the first embodiment. However, as alsodescribed above, the description for j is omitted in order to simplifythe explanation.

The processing of the step S33 is similar to the processing of the stepS13 in the first embodiment. Therefore, the fitting curve y(t) isrepresented as follows:

$\begin{matrix}{{y(t)} = {{\frac{1}{{2n} + 1}a_{0}} + {\frac{2}{{2n} + 1}{\sum\limits_{q = 1}^{n}\left( {{a_{q}\cos \frac{2q\; \pi}{{2n} + 1}t} + {b_{q}\sin \frac{2q\; \pi}{{2n} + 1}t}} \right)}}}} & (11)\end{matrix}$

The meanings of the variables and coefficients in the expression (11)are same as those in the first embodiment. And, as explained in thefirst embodiment, the right side of the expression (11) is representedby a sum of the exponential functions as follows:

$\begin{matrix}{{y(t)} = {\sum\limits_{q = {- n}}^{n}{c_{q}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}}} & (12)\end{matrix}$

Then, the first fitting curve generator 331 performs differential orintegral of the fitting curve y(t), which is represented by the sum ofthe exponential functions, to generate a curve u(t) (step S35). When thetype of data is not changed, the differential is performed whileassuming p=0.

Although the curve u(t) is represented like the expression (10), thecurve u(t) is represented as follows when considering the expression(12).

$\begin{matrix}{{{u(t)} = {{\frac{^{p}}{t^{p}}{y(t)}} = {\sum\limits_{q = {- n}}^{n}{\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)^{p}c_{q}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}}}}\begin{matrix}{{u(t)} = {\int{{y(t)}{t}}}} \\{= {{\sum\limits_{q = {- n}}^{- 1}{\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)^{- 1}c_{q}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}} + {c_{0}t} +}} \\{{\sum\limits_{q = 1}^{n}{\left( \frac{2\sqrt{- 1}q\; \pi}{{2n} + 1} \right)^{- 1}c_{q}^{\frac{2\sqrt{- 1}q\; \pi}{{2n} + 1}t}}}}\end{matrix}} & (13)\end{matrix}$

As described above, y(t) is the sum of the exponential functions.Therefore, in case of the first-order differential, its coefficientbecomes a coefficient of the exponent of the exponential function, incase of the second-order differential, its coefficient becomes (thecoefficient of the exponent of the exponential function)², in case ofthe third-order differential, its coefficient becomes (the coefficientof the exponent of the exponential function)³, and in case of them-order differential, its coefficient becomes (the coefficient of theexponent of the exponential function)^(m). Thus, the differentialoperation becomes very easy. Moreover, the higher-order differential ispossible, although it is impossible in case of the spline method.Furthermore, typically, the integration of the exponential functione^(kx) becomes e^(kx)/k+C (k is a coefficient of the exponent and C isan integral constant), and the calculation is very easy. Therefore, itis also possible to perform the integral of the y(t) plural times.

For example, when y(t) is a fitting curve for the velocity, a curve u(t)for the acceleration is obtained by performing the first-orderdifferential, and a curve u(t) for the distance is obtained byperforming the integral. Moreover, when y(t) is the fitting curve forthe distance, a curve u(t) for the velocity is obtained by performingthe first-order differential, and a curve u(t) for the acceleration isobtained by performing the second-order differential.

Then, the first fitting curve generator 331 calculates data of 2m+1discrete points u₀, u₁, . . . , u_(2m) for the curve u(t) generated atthe step S35 (step S37), and stores the calculated data in theinput/output data storage unit 320. Then, the processing ends.

Because 2m+1 discrete points are disposed in a section from t=0 to t=2n,data of the 2m+1 discrete points u₀, u₁, . . . , u_(2m) are as follows:

${u_{0} = {u(0)}},{u_{1} = {u\left( \frac{n}{m} \right)}},{u_{2} = {u\left( \frac{2n}{m} \right)}},\ldots \mspace{14mu},{u_{{2m} - 1} = {u\left( \frac{n\left( {{2m} - 1} \right)}{m} \right)}},{u_{2m} = {u\left( {2n} \right)}}$

The first fitting curve generator 331 can perform the systemidentification, which was explained in the first embodiment, by usingthe data of the 2m+1 discrete points u₀, u₁, . . . , u_(2m), which arestored in the input/output data storage unit 320, and data of the 2m+1discrete points v₀, v₁, . . . , v_(2m).

When the aforementioned processing is performed, y(t) and u(t) arecontinuous functions. Therefore, it becomes possible to equate thesampling intervals for the digital inputs with the sampling intervalsfor the digital outputs. Moreover, because the differentiation (however,p≠0) and integration of the fitting curve y(t) can be easily performed,it becomes possible to perform the system identification, analysis fordata having a different type.

Although the embodiments of this invention were explained, thisinvention is not limited to those. For example, the explanation was madein which the accelerometer 101 and the distance sensor 102, which areprovided in the vehicle 1000, are presumed, however, data measured byother sensors that correspond to inputs and outputs of a system to beidentified may be employed.

Moreover, functional block diagrams illustrated in FIGS. 3 and 4 aremere examples, and do not correspond to program module configurations.Furthermore, as for the processing flow, as long as the processingresults do not change, turns of the steps may be exchanged, and pluralsteps may be executed in parallel.

Furthermore, in FIGS. 3 and 4, the detection apparatus 100 and theinformation processing apparatus 300 are separately implemented,however, an apparatus that integrates them may be implemented dependingon the usage. Furthermore, the detection apparatus 100 may beimplemented in an apparatus such as a cellular phone or car navigationapparatus.

Furthermore, the embodiments can be applied not only to the systemidentification, but also to a learning problem of the artificialintelligence.

In addition, the aforementioned information processing apparatus 300 isa computer device as shown in FIG. 13. That is, a memory 2501 (storagedevice), a CPU 2503 (processor), a hard disk drive (HDD) 2505, a displaycontroller 2507 connected to a display device 2509, a drive device 2513for a removable disk 2511, an input unit 2515, and a communicationcontroller 2517 for connection with a network are connected through abus 2519 as shown in FIG. 13. An operating system (OS) and anapplication program for carrying out the foregoing processing in theembodiment, are stored in the HDD 2505, and when executed by the CPU2503, they are read out from the HDD 2505 to the memory 2501. As theneed arises, the CPU 2503 controls the display controller 2507, thecommunication controller 2517, and the drive device 2513, and causesthem to perform necessary operations. Besides, intermediate processingdata is stored in the memory 2501, and if necessary, it is stored in theHDD 2505. In this embodiment of this technique, the application programto realize the aforementioned functions is stored in thecomputer-readable, non-transitory removable disk 2511 and distributed,and then it is installed into the HDD 2505 from the drive device 2513.It may be installed into the HDD 2505 via the network such as theInternet and the communication controller 2517. In the computer asstated above, the hardware such as the CPU 2503 and the memory 2501, theOS and the necessary application programs systematically cooperate witheach other, so that various functions as described above in details arerealized.

The aforementioned embodiments are outlined as follows:

An information processing method relating to the embodiments includes(A) generating a fitting curve represented by a sum of exponentialfunctions for each of a set of digital inputs and a set of digitaloutputs for a physical system that is represented by one or pluralequations including m-order differential operators (m is an integerequal to or greater than 1. The set of the digital inputs and the set ofthe digital outputs are stored in a data storage unit.); and (B)calculating coefficients of the differential operators, which areincluded in first coefficients, so that a first coefficient of eachexponential function included in an expression obtained by a product ofthe differential operators and the fitting curve for the set of thedigital inputs is equal to a second coefficient of the same exponentialfunction, which is included in the fitting curve for the set of thedigital outputs.

When the fitting curve is represented by a sum of exponential functionsas described above, it is possible to easily differentiate the fittingcurve even if m is higher-order, and it is also possible to easilyobtain the coefficients of the differential operators. In other words, acalculation amount can be reduced as a whole, and it is possible toperform the system identification.

The aforementioned calculating may include: (b1) generating pluralcombinations of a third coefficient and a difference between the secondcoefficient and a constant term, for which a coefficient of thedifferential operators is not multiplied, among the first coefficients,wherein the third coefficient is a coefficient, for which a coefficientof the differential operators is multiplied, among the firstcoefficients; and (b2) performing linear regression calculation for theplural combinations to calculate the coefficients of the differentialoperators.

Accordingly, even when a lot of digital inputs and digital outputs areobtained, it is possible to obtain the coefficients of the differentialoperators with much less error.

Moreover, the aforementioned first coefficient may be a product of avalue of the differential operator for a coefficient in an exponent of afirst exponential function corresponding to the first coefficient and acoefficient of the same exponential function as the first exponentialfunction, which is included in the fitting curve for the set of thedigital inputs. The differential can be represented by such simpleoperations.

Furthermore, the aforementioned information processing method mayfurther include: (C) generating a second fitting curve represented by asum of exponential functions for a second set of digital inputs of thephysical system; (D) calculating, for each exponential function of thesecond fitting curve, coefficients of an output function, which areobtained from a coefficient of the exponential function and a value of adifferential operator, which includes the calculated coefficient of thedifferential operator for a coefficient in an exponent of theexponential function; and (E) calculating digital output values from theoutput function including the calculated coefficient of the outputfunction. By performing the aforementioned processing, it becomespossible to predict the outputs of the identified system from thedigital inputs.

Moreover, the aforementioned generating may include: (a1) generating thefitting curve for the set of the digital inputs by calculating n+1frequencies by performing discrete Fourier transformation for 2n+1digital inputs and generating n-order Fourier series from the n+1frequencies, wherein n is an integer; and (a2) generating the fittingcurve for the set of the digital outputs by calculating n+1 frequenciesby performing discrete Fourier transformation for 2n+1 digital outputsand generating n-order Fourier series from the n+1 frequencies. Byperforming the aforementioned processing, it becomes possible togenerate the fitting curve that can recover the original digital inputsand outputs, securely.

Moreover, the aforementioned generating may further include: (a3)calculating l+1 frequencies (l is an integer.) by performing discreteFourier transformation for 2l+1 third digital inputs that are stored inthe data storage unit, and generating 1-order Fourier series from thel+1 frequencies to generate a third fitting curve represented by a sumof exponential functions for the third digital inputs; and (a4)calculating 2n+1 digital inputs from the generated third fitting curve,and storing the 2n+1 digital inputs into the data storage unit. When theaforementioned processing is performed, it is possible to perform thesystem identification even if the number of digital inputs stored in thedata storage unit and the number of digital outputs are different eachother.

Moreover, the aforementioned generating may include: (a5) generating afirst curve by performing differential or integral for the fitting curvefor the set of the digital inputs; and (a6) generating fourth digitalinputs by using the generated first curve, and storing the fourthdigital inputs into the data storage unit. Because the fitting curve forthe set of digital inputs are represented by the sum of the exponentialfunctions, it is possible to easily calculate the differential andintegral. Then, when performing the aforementioned processing, itbecomes possible to perform the system identification for data obtainedby the differential or integral.

Incidentally, it is possible to create a program causing a computer toexecute the aforementioned processing, and such a program is stored in acomputer readable storage medium or storage device such as a flexibledisk, CD-ROM, DVD-ROM, magneto-optic disk, a semiconductor memory suchas ROM (Read Only Memory), and hard disk. In addition, the intermediateprocessing result is temporarily stored in a storage device such as amain memory or the like.

All examples and conditional language recited herein are intended forpedagogical purposes to aid the reader in understanding the inventionand the concepts contributed by the inventor to furthering the art, andare to be construed as being without limitation to such specificallyrecited examples and conditions, nor does the organization of suchexamples in the specification relate to a showing of the superiority andinferiority of the invention. Although the embodiments of the presentinventions have been described in detail, it should be understood thatthe various changes, substitutions, and alterations could be made heretowithout departing from the spirit and scope of the invention.

What is claimed is:
 1. An information processing apparatus, comprising:a memory; and a processor configured to use the memory and execute aprocess, the process comprising: generating a fitting curve representedby a sum of exponential functions for each of a set of digital inputsand a set of digital outputs for a physical system that is representedby one or a plurality of equations including m-order differentialoperators, wherein m is an integer equal to or greater than 1; andcalculating coefficients of the differential operators, which areincluded in first coefficients, so that a first coefficient of eachexponential function included in an expression obtained by a product ofthe differential operators and the fitting curve for the set of thedigital inputs is equal to a second coefficient of the same exponentialfunction, which is included in the fitting curve for the set of thedigital outputs.
 2. The information processing apparatus as set forth inclaim 1, wherein the calculating comprises: generating a plurality ofcombinations of a third coefficient and a difference between the secondcoefficient and a constant term, for which a coefficient of thedifferential operators is not multiplied, among the first coefficients,wherein the third coefficient is a coefficient, for which a coefficientof the differential operators is multiplied, among the firstcoefficients; and performing linear regression calculation for theplurality of combinations to calculate the coefficients of thedifferential operators.
 3. The information processing apparatus as setforth in claim 1, wherein the first coefficient is a product of a valueof the differential operator for a coefficient in an exponent of a firstexponential function corresponding to the first coefficient and acoefficient of the same exponential function as the first exponentialfunction, which is included in the fitting curve for the set of thedigital inputs.
 4. The information processing apparatus as set forth inclaim 1, wherein the process further comprising: generating a secondfitting curve represented by a sum of exponential functions for a secondset of digital inputs of the physical system; calculating, for eachexponential function of the second fitting curve, coefficients of anoutput function, which are obtained from a coefficient of theexponential function and a value of a differential operator, whichincludes the calculated coefficient of the differential operator for acoefficient in an exponent of the exponential function; and calculatingdigital output values from the output function including the calculatedcoefficient of the output function.
 5. The information processingapparatus asset forth in claim 1, wherein the generating comprises:generating the fitting curve for the set of the digital inputs bycalculating n+1 frequencies by performing discrete Fouriertransformation for 2n+1 digital inputs and generating n-order Fourierseries from the n+1 frequencies, wherein n is an integer; and generatingthe fitting curve for the set of the digital outputs by calculating n+1frequencies by performing discrete Fourier transformation for 2n+1digital outputs and generating n-order Fourier series from the n+1frequencies.
 6. The information processing apparatus as set forth inclaim 5, wherein the generating further comprises: calculating l+1frequencies by performing discrete Fourier transformation for 2l+1 thirddigital inputs, and generating l-order Fourier series from the l+1frequencies to generate a third fitting curve represented by a sum ofexponential functions for the third digital inputs; and calculating 2n+1digital inputs from the generated third fitting curve.
 7. Theinformation processing apparatus as set forth in claim 1, wherein thegenerating comprises: generating a first curve by performingdifferential or integral for the fitting curve for the set of thedigital inputs; and generating fourth digital inputs by using thegenerated first curve.
 8. A non-transitory computer-readable storagemedium storing a program for causing a computer to execute a process,the process comprising: generating a fitting curve represented by a sumof exponential functions for each of a set of digital inputs and a setof digital outputs for a physical system that is represented by one or aplurality of equations including m-order differential operators, whereinm is an integer equal to or greater than 1; and calculating coefficientsof the differential operators, which are included in first coefficients,so that a first coefficient of each exponential function included in anexpression obtained by a product of the differential operators and thefitting curve for the set of the digital inputs is equal to a secondcoefficient of the same exponential function, which is included in thefitting curve for the set of the digital outputs.
 9. An informationprocessing method, comprising: generating, by using a computer, afitting curve represented by a sum of exponential functions for each ofa set of digital inputs and a set of digital outputs for a physicalsystem that is represented by one or a plurality of equations includingm-order differential operators, wherein m is an integer equal to orgreater than 1; and calculating, by using the computer, coefficients ofthe differential operators, which are included in first coefficients, sothat a first coefficient of each exponential function included in anexpression obtained by a product of the differential operators and thefitting curve for the set of the digital inputs is equal to a secondcoefficient of the same exponential function, which is included in thefitting curve for the set of the digital outputs.